Today we continued the project and calculated the two probabilities in the “paradox.” These calculations are pretty challenging ones for kids, but even with the counting challenges, this was a really fun project.

I started by reminding them of the problem and getting their thoughts from yesterday:

Now we calculated the probability of having a second ace given that you have at least one ace. It took a while to find the right counting ideas, but once they did the calculation went pretty quickly. The counting technique that we used here was case by case counting:

Next we moved to the 2nd problem -> If you have the Ace of Spades, what is the probability that you have more than one ace? The counting technique that we used here was complimentary counting:

Finally, I asked the boys to reflect on the problem – was it still a “paradox” in their minds or did it make a bit more sense now that we worked through it?

I really loved talking through this problem with the boys – thanks to John Cook for sharing it and to Futility Closet for writing about it originally!

My reaction was that it would be fun to turn this into a project for kids, but this one would need a little introduction since conditional probability can be incredibly non-intuitive. During the week I came up with a plan, and we began to look at the problem this morning.

Here’s the introduction – I asked the boys to give their initial reaction to the seeming paradox:

Next we looked at an example that is slightly easier to digest -> rolling two dice and asking “do you have at least one 6?”

My younger son had a little trouble with the conditional probability, so I’m happy that we took this introductory path:

Next we moved to a slightly more difficult problem -> rolling 3 distinct dice. I used a 6x6x6 Rubik’s cube to represent the 216 states. To start, I asked the boys to count the number of states that had at least one six. Their approach to counting those 91 states was really fascinating:

Finally, we looked at the analogy to the 2nd ace paradox in our setting. So, if you have “at least one 6” what is the chance that you have more than one six, and if you have “a six on a specific die” what is the chance that you have more than one six?

Again, my younger son had a little trouble understanding how the cube represented the various rolls, but being able to hold the cube and see the states helped him get past that trouble:

Tomorrow we’ll move on to studying the paradox with the playing cards. Hopefully today’s introduction helped the boys understand

The idea is easy to play with on your own -> deal out a standard deck of cards (arranged in any order you like) into 13 piles of 4 cards. By picking any card you like (but exactly one card) from each of the 4 piles, can you get a complete 13-card sequence Ace, 2, 3, . . . , Queen, King?

Here’s how I introduced Wright’s puzzle. I started the way he started – when you deal the 13 piles, is it likely that the top card in each pile will form the Ace through King sequence:

Now we moved on to the main problem – can you choose 1 card from each of the 13 piles to get the Ace through King sequence?

As always, it is fascinating to hear how kids think through advanced mathematical ideas. By the end of the discussion here both kids thought that you’d always be able to rearrange the cards to get the right sequence.

Now I had the boys try to find the sequence. Their approach was essentially the so-called “greedy algorithm”. And it worked just fine.

To wrap up, we shuffled the cards again and tried the puzzle a second time. This time it was significantly more difficult to find the Ace through King sequences, but they got there eventually.

They had a few ideas about why their procedure worked, but they both thought that it would be pretty hard to prove that it worked all the time.

I’m always happy to learn about advanced math ideas that are relatively easy to share with kids. Wright’s card puzzle is one that I hope many people see and play around with – it is an amazing idea for kids (and everyone!) to see.

We stumbled on this problem in the book my older son is studying over the summer:

A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.

(i) What is the probability of finishing the game with a positive score?

(ii) What is the expected win when you play this game?

The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.

Here’s how the conversation went:

(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.

(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.

(3) Now that we made progress on one of the tough cases, we tackled the other two:

(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.

(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.

I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.

I remember this problem from way back when I was studying for the AHSME back in the mid 1980s. I thought it would be fun to talk through this problem with my older son – it has some great lessons. One lesson in particular is that there is a difference between counting paths and calculating probabilities. It was most likely this problem that taught me that lesson 30+ years ago!

So, here’s my son’s initial reaction to the problem:

Next we talked through how to calculate the probabilities. This calculation gave him more trouble than I was expecting. He really was searching for a rule for the probabilities that would work in all situations – but the situations are different depending on where you are in the grid!

Despite the difficulty, I’m glad we talked through the problem.

(also, sorry about the phone ringing in the middle of the video!)

So, definitely a challenging problem, but also a good one to help kids begin to understand some ideas about probability.

One of the topics covered in the essay is a special type of card shuffle called the Faro shuffle. We have done a few projects on card shuffling projects previously, so I thought the kids would be interested in learning about the Faro shuffle. Here are our prior card shuffling projects:

I started the project by asking the kids what they knew about cards. They remembered some of the shuffling projects and then introducing the idea of the Faro shuffle.

My younger son thought he saw a connection with pi, which was a fun surprise.

We continued studying the Faro shuffle with 8 cards and looked for patterns in the card numbers and positions. The boys noticed some neat patterns and were able to predict when we’d return to the original order of cards!

Next we looked at the paths taken by individual cards. My older son thought that there might be a connection with modular arithmetic (!!!) and the boys were able to find the pattern. I’d hoped that finding the pattern here would be within their reach, so it was a really nice moment when he brought up modular arithmetic.

Finally, we wrapped up by talking about how to extend the ideas to a 52 card deck and calculated how many Faro shuffles we’d need to get back to where we started.

I think that kids will find the idea of the Faro shuffle to be fascinating. Simply exploring the number patterns is a really interesting project, and there’s lots of really interesting math connected to the idea. I’m really thankful that Jim Propp takes the time to produce these incredible essays each month. They are a fantastic (and accessible) way to explore lots of fun mathematical ideas.

By coincidence I heard the recent Ben Ben Blue podcast yesterday which had a brief mention / lament that it was hard to share mistakes in videos.

This problem is probably a good challenge problem for my older son and definitely above the level of my younger son. But listening to both of them try to work through the problem was really interesting.

I started with my older son – he initially approached the problem by comparing the individual probabilities:

After his initial work, I talked with him about comparing the probabilities of the complete events described in the problem. Initially there was a little confusion on his part, but eventually he understood the idea:

Next up was my younger son – not surprisingly, he had a hard time getting started with the problem. His initial approach was similar to what my older son had done – he looked at the one head and two heads events separately to see which one was more likely for each coin:

As I did with my older son, I asked him to look at the two events as a single event and see which one was more likely when each coin went first:

So, a nice project and an opportunity to see a few mistakes and as well as how kids approach a challenging probability problem.